Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs
Abstract
:1. Introduction
2. Non-Isolated Resolving Set of Where Is a -Regular Graph
3. Non-Isolated Resolving Set of Where Is a -Regular Graph
- 1.
- Let ;
- 2.
- Let ;
- 3.
- Choose two non-adjacent vertices in , say and and say another unlabeled vertex that is not adjacent to as ;
- 4.
- Let ;
- 5.
- Let be a vertex that is not adjacent to ;
- 6.
- If , then , and define , then repeat Step 5. Otherwise, go to Step 7;
- 7.
- Let be an induced subgraph of by ;
- 8.
- Define ;
- 9.
- If , then define , and repeat Step 3. Otherwise, finish.
- (i)
- If , then W is an -set of ;
- (ii)
- If , then .
- (i)
- Since , we have . Hence, W is also an -set of ;
- (ii)
- Since is a -regular graph, by Lemma 1, there is such that where and . Let W be an -set of and for every . Next, it is proven by contradiction. Suppose there is such that . As a consequence, there are two vertices u and v in such that . Since every is adjacent to all vertices in , we have . We obtain a contradiction.
3.1. Resolvability of a Complete Graph Minus a Hamiltonian Cycle
- (i)
- Every gap of W contains at most three vertices;
- (ii)
- At most one gap of W contains three vertices;
- (iii)
- If a gap of W contains at least two vertices, then its neighboring gaps contain at most one vertex.
- (i)
- Suppose there is a gap of W containing four vertices for some . We obtain . We obtain a contradiction. Hence, every gap of W contains at most three vertices;
- (ii)
- Suppose there are two different gaps containing three vertices and for some k and l in with . We obtain . This is a contradiction. Therefore, the number of gaps of W containing three vertices is at most one;
- (iii)
- Suppose there are five vertices for some such that is the only vertex of W. Since and for every , we obtain . This is a contradiction. We conclude that if A is a gap of W containing at least two vertices, then the neighboring gaps of A contain at most one vertex.
- v belongs to a gap of size one of W:Let and be the end points of a gap containing v. Then, the vertex v has a distance of two to both and . Since , for every , we have or . This implies ;
- v belongs to a gap of size two of W:Let and be the end points of a gap containing v. Without loss of generality, let , then and . If there is with and and , then there is such that and . Therefore, for every , we obtain
- v belongs to a gap of size three of W:Let and be the end points of a gap containing v. Let , then . By Properties and of Lemma 3, no other vertex of has this representation. Now, let or . Without loss of generality, let , then and . If there is with , and , then there is such that and . Therefore, for every , we obtain .
- for some :The number of vertices in all gaps of W is at most . Therefore, . This implies that ;
- for some :The number of vertices in all gaps of W is at most . Therefore, . This means that .
- :Therefore, for some . Thus, . We define . Note that W contains vertices and satisfies Properties of Lemma 3;
- :Therefore, for some . Thus, . For , we define . For , we define . Note that W contains vertices and satisfies Properties of Lemma 3;
- :Therefore, for some . Thus, . We define . Note that W contains vertices and satisfies Properties of Lemma 3;
- :Therefore, for some . Thus, . For , we define . For , we define . Note that W contains vertices and satisfies Properties of Lemma 3;
- :Therefore, for some . Thus, . We define . Note that W contains vertices and satisfies Properties of Lemma 3.
3.2. -Set of a -Regular Graph
- (i)
- Every gap of W contains at most three vertices;
- (ii)
- At most one gap of W contains three vertices.
- (i)
- Suppose there is a gap of W containing four vertices for some and . We obtain . We obtain a contradiction. Hence, every gap of W contains at most three vertices;
- (ii)
- By considering Lemma 4, suppose there are two different gaps containing three vertices in of and in of for some and with . We obtain . We have a contradiction. Hence, the number of gaps of W containing three vertices is at most one.
- For , we have and ;
- For , we have , and .
3.3. -Set of Where Is a -Regular Graph
- If , then there exist two vertices x and y in such that . This implies that , a contradiction;
- If , then contains an isolated vertex, a contradiction.
- x and y in for some :Let and . Since , we obtain ;
- and for some j and k in with :Let and for some a and b in . Since , we obtain .
- If , then there exist two vertices x and y in such that . This implies that , a contradiction;
- For , if contains an isolated vertex, then we have a contradiction. Otherwise, then there exist two vertices x and y in such that for every . This implies that , a contradiction.
- x and y in for some :Since is an -set of , we have . This means that ;
- and for some :If , then for some with and for any , we have . If , then for any , we have . Therefore, , and this means that ;
- and for some j and k in with :For any , we have but . Therefore, , so that ;
- x and y in :Let and for some j and k in with . For any , we have . Therefore . This means that .
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Abidin, W.; Salman, A.; Saputro, S.W. Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs. Mathematics 2022, 10, 962. https://doi.org/10.3390/math10060962
Abidin W, Salman A, Saputro SW. Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs. Mathematics. 2022; 10(6):962. https://doi.org/10.3390/math10060962
Chicago/Turabian StyleAbidin, Wahyuni, Anm Salman, and Suhadi Wido Saputro. 2022. "Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs" Mathematics 10, no. 6: 962. https://doi.org/10.3390/math10060962
APA StyleAbidin, W., Salman, A., & Saputro, S. W. (2022). Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs. Mathematics, 10(6), 962. https://doi.org/10.3390/math10060962